Slides & Videos

Abstract: Macdonald polynomials appear in the study of p-adic Whittaker functions in intriguing ways. For example, the infinite dimensional analogue of the Casselman-Shalika formula gives rise to a correction factor. In the affine type, this factor is known explicitly by Cherednik’s work on Macdonald’s constant term conjecture. A similar factor appears when one extends other formulas from the theory of p-adic groups to the affine, and general Kac-Moody setting. For example, Macdonald’s formula for the spherical function shares this feature. We will discuss this correction factor, and how methods of computing imaginary root multiplicities, such as the Peterson algorithm and the Berman-Moody formula can be generalised to compute it. We will also mention the relationship between constructions of Iwahori-Whittaker functions and nonsymmetric Macdonald polynomials. In the classical setting, this is understood by the work of Brubaker, Bump and Licata. More recently, work of Sahi, Stokman and Venkateswaran indicates that a similar connection exists in the metaplectic context. The talk will cover joint results with Dinakar Muthiah, Manish Patnaik, and Ian Whitehead.

Abstract: Amazing nabla operator discovered by Bergeron and Garsia is a cornerstone of the theory of Macdonald polynomials. Applying it to various symmetric functions produces interesting generating functions of Dyck paths and parking functions. These kind of results are sometimes known as "shuffle theorems". I will try to give an overview of these results and explain how working with affine permutations and certain generalized P-tableaux allows to view them from a uniform point of view. The "new" in the title refers to the formula conjectured by Loehr and Warrington giving an explicit expansion of nabla of a Schur function in terms of nested Dyck paths.

This is a joint work with Erik Carlsson.

Abstract: LLT polynomials are a family of symmetric functions introduced by Lascoux, Leclerc and Thibon in 1997 which naturally arise in the description of the power-sum plethysm operators on symmetric functions. Grojnowski and Haiman proved that they are Schur positive using Kazhdan-Lusztig theory, but there is no known combinatorial formula for the Schur coefficients. Due to the discovery of local linear relation by Seungjin Lee in 2018, there has been a progress on figuring out combinatorial formulas for Schur coefficients in certain special cases in recent several years. In the first part of this talk, we explain some known results in the case of certain subfamilies of LLT diagrams. In the second part, we utilize linear relations introduced by Lee to prove some combinatorial formulas for the Schur coefficients of LLT polynomials, when the LLT polynomials are indexed by certain diagrams related to particular type of graphs. This is joint work with Jisun Huh and Sun-Young Nam.

Abstract: I will explain the structure of natural endofunctors of the category of modules over current algebras, whose derived version satisfies the relations of generators of DAHA. As a byproduct of this construction, we obtain certain complexes whose Euler characteristic is equal to Macdonald polynomials. Fortunately, in the case of sl_2, these complexes are modules.

Abstract: We will review various links between the characters of modules of current algebras and nonsymmetric Macdonald polynomials. We will start with recalling the theorem due to Sanderson and Ion, which computes the characters of the affine Demazure modules as t=0 specializations of the nonsymmetric Macdonald polynomials. We will continue with the specialization at infinity and the generalized Weyl modules, describing certain constructions due to Feigin, Cherednik, Orr, Makedonskyi and Kato. Finally, we will mention the recent work by Kato, Khoroshkin and Makedonskyi, where a categorification of the non specialized Macdonald polynomials is presented.

Abstract: Macdonald polynomials are a remarkable family of functions. They are a common generalization of many different families of special functions arising in the representation theory of reductive groups, including spherical functions and Whittaker functions.

In turn, Macdonald polynomials can be understood in terms of a certain representation of Cherednik's double affine Hecke algebra (DAHA), acting on polynomial functions on a torus.

Whittaker functions admit a natural generalization to the setting of metaplectic covers of reductive p-adic groups, which play a key role in the theory of Weyl group multiple Dirichlet series.

It turns out that Macdonald polynomials also admit a corresponding generalization, which can be understood in terms of a representation of the DAHA on the space of quasi-polynomial functions on a torus. This is joint work with Jasper Stokman and Vidya Venkateswaran.

Abstract: This talk will be about a recent formulation of the Lascoux--Leclerc--Thibon (LLT) polynomials, using integrable rank-n vertex models. A number of known properties of the LLT polynomials may be deduced from this formulation, including their reduction to modified Hall--Littlewood polynomials for sufficiently large n. I will present a new combinatorial formula for the expansion coefficients of the LLT polynomials over the modified Hall--Littlewood basis; this formula may be viewed as (q,n)-generalization of Knutson--Tao--Woodward honeycombs for the Littlewood--Richardson coefficients. As is known from work of Haglund--Haiman--Loehr, summing LLT polynomials over ribbon diagrams leads to the modified Macdonald polynomials; carrying out this procedure on the underlying honeycombs and making various special choices of the parameters leads to new combinatorial formulae for the Kostka--Foulkes polynomials and the expansion coefficients of Jack polynomials over the monomial basis. This talk is based on the joint work https://arxiv.org/abs/2101.01605 with Amol Aggarwal and Alexei Borodin.

Abstract: In this talk I will review the computation of the stationary measure of one dimensional local integrable stochastic processes through exchange/reflection equations, leading typically to consider polynomial representations of affine Hecke algebras. I shall focus mainly on the case of interacting particle systems, namely multispecies variant of the (Totally) Asymmetric Simple Exclusion process, for which, depending on the boundary conditions and on the rates of the process, the results are expressed in terms of Macdonald, Koornwinder or Schubert polynomials.

Abstract: The fact that the modular group arises as a group of outer automorphisms of certain double affine Hecke algebras was discovered by Cherednik. One of the initial applications was Cherednik's definition of the difference Fourier transform (which is given by the action of a particular element in the modular group) which is the key structure that explains the phenomenon behind Macdonald's evaluation-duality conjecture. Other applications followed, to the harmonic analysis in the context of double affine Hecke algebras, to Rogers-Ramanujan type identities for affine root systems, and to the topology of torus knots.

In the most general construction, the double affine Hecke algebras are attached to double reductive group data. From this point of view, Cherednik's result covers only the algebras attached to untwisted simply connected data. I will report on my joint work with S. Sahi where we explain that all double affine Hecke algebras admit a congruence subgroup of the modular group acting as outer automorphisms. The full description of this congruence subgroup (e.g. the level) can be explicitly determined from the symmetries of the double reductive group data. This points to a potentially new source of modularity arising from the representation theory of the double affine Hecke algebras. The result is a consequence of the description of double Hecke algebras attached to double data of adjoint type in terms of certain crystallographic diagrams (double affine Coxeter diagrams).

Abstract: I will review the connection between Macdonald polynomials and quantum-integrable spin chains with long-range interactions. The main character is the q-deformed (XXZ-like) Haldane--Shastry spin chain, which was found by Uglov long ago in a little-known preprint to be rediscovered and more thoroughly understood in recent years. The many remarkable properties of this spin chain include quantum-loop symmetry and exact eigenvectors featuring (the quantum zonal spherical case of) Macdonald polynomials. I will explain how the underlying algebraic structure, the affine Hecke algebra, can be used to construct a spin-version of Macdonald theory via quantum-affine Schur--Weyl duality, and how this reduces to a long-range spin chain by a procedure known as 'freezing'.

If time allows I will outline our diagonalisation. Here the key step is to show that on a subspace of suitable polynomials, the first few 'classical' (i.e. no difference part) Y-operators reduce, when the variables are specialised to roots of unity, to Y-operators in less variables with parameters at the quantum zonal spherical point. This is based on joint work with Vincent Pasquier and Didina Serban, and work in progress.

Abstract: We review some connections between Macdonald polynomials and multivariate basic hypergeometric series. It is well-known that Macdonald polynomials are multi-series extensions of the continuous $q$-ultraspherical polynomials, which are a family of orthogonal polynomials that admit a representation in terms of well-poised $_2\phi_1$ basic hypergeometric series. Less known is the fact that the Pieri formula for Macdonald polynomials (originally conjectured by Macdonald and first proved by Koornwinder) can be viewed as a multi-series extension of the $q$-Pfaff--Saalsch\"utz summation, which is a summation for a terminating balanced $_3\phi_2$ series. The inverse of the Pieri formula was established by Michel Lassalle and the speaker and can be viewed as a multi-series extension of the terminating very-well-poised $_6\phi_5$ summation. The underlying matrix inversion actually forms a bridge between identities seemingly related to Macdonald polynomials (the respective multiple series contain a Macdonald determinant as a factor) and identities of Gustafson--Milne type (with a Vandermonde determinant as a factor of the respective multiple series). We make some of these connections explicit and show a couple of identities (of series containing a Macdonald determinant as a factor) that we obtained by applying inverse relations. Our results mainly concern series that can be associated to the root system $A_n$; however one of our formal results includes a $C_n$ $_8\phi_7$ summation that is still conjectural. If time permits we will turn to some multivariate extensions of Ramanujan's $_1\psi_1$ summation that we deduced from the aforementioned results.

Abstract: SSV polynomials are a new generalization of Macdonald polynomials discovered by Sahi, Stokman, and Venkateswaran, unifying the theory of Macdonald polynomials with that of Iwahori Whittaker functions on metaplectic covers of reductive p-adic groups. Like both Macdonald polynomials and Whittaker functions, SSV polynomials satisfy a recursion coming from a representation of the corresponding double affine Hecke algebra. We will use this recursion to give a combinatorial formula for SSV polynomials in terms of alcove walks, generalizing Ram and Yip's formula for Macdonald polynomials. We will then discuss several applications, including an alcove walk formula for metaplectic Iwahori Whittaker functions.

Abstract: In this talk I will describe a number of strange conjectures in Macdonald-Koornwinder theory that have arisen in recent joint work with Chul-hee Lee and Eric Rains on the branching problem.

Abstract: Recently, a formula for the symmetric Macdonald polynomials was given by Corteel, Mandelshtam and Williams in terms of objects called multiline queues, which also compute probabilities of a statistical mechanics model called the multi species ASEP on a ring. It is natural to ask whether the modified Macdonald polynomials can be obtained using a combinatorial gadget for some other statistical mechanics model. We answer this question in the affirmative by giving a new formula for the modified Macdonald polynomials in terms of fillings of tableaux that we call polyqueue tableaux. We show that these tableaux encode stationary probabilities for a multi species Totally Asymmetric Zero Range Processs (TAZRP) on a ring with parameter q=1. This is joint work with J. Martin and O. Mandelshtam (arXiv:2011.06117).